The derivative operator [tex]\frac{d}{dn}[/tex] should operate on the sum.
I am not even sure what it means to have delta t inside a sum where t is a dumby variable, and then taking a limit of it as it approaches zero, the notation here is quite problematic.

I think I know what happened. It must have again considered d as constant rather than an infinitesimal.There seems to be a simpler solution using the second fundamental theorem of calculus and that would yield